Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 21F (Algebraic Topology)

Spaces $X$ and $Y$ are homotopy equivalent if there are continuous mapssuch thatThe maps $f$ and $g$ are called homotopy inverses.

A subspace $Y\subseteq X$ is a retract of $X$ if there is a continuous retractionsuch that $r(y)=y$ for every $y\in Y$. A space is a contractible space if its identity map is homotopic to a constant map.

Let $i:Y\hookrightarrow X$ be the inclusion and letbe a contraction, with $H(x,0)=x$ and $H(x,1)=x_0$. Thenis continuous and satisfiesThus $K$ contracts $Y$, proving that every retract of a contractible space is contractible.

Choose a contractionApply the homotopy extension property to the initial map ${\mathrm{id}}_X:X\to X$ and to $h$, regarded as a homotopy into $X$. It suppliessuch that

At $t=1$, the map $H_1$ sends every point of $A$ to $a_0$. It is therefore constant on the equivalence classes of the quotient topology $X/A$. By the universal property of the quotient topology, there is a continuous mapsuch thatwhere $q:X\to X/A$ is the quotient map. The homotopy $H$ immediately gives

For every $t$, the map $q\circ H_t$ is constant on $A$, because $H(a,t)=h(a,t)\in A$. It therefore descends toThis descended map is continuous: the product $q\times {\mathrm{id}}_I$ is a quotient map because $I$ is compact Hausdorff, and $q\circ H$ is constant on its fibres.
At the endpoints,ThusThe maps $q$ and $g$ are homotopy inverses, soThis proves the theorem on collapsing a contractible cofibration.